Inverse problem of scattering and off-shell continuation of the transition amplitude

It is shown how dispersion relations for the Jost solution can be used to generate a class of integral equations, among them the Marchenko equation, that all solve the inverse problem for the $s$-wave Schr\"odinger equation. Motivated by the fact that scattering data (the phase shifts) are functions of momentum, we develop momentum-space versions of some of these equations. We also construct a Fredholm series solution to the inverse problem; it is formulated entirely in momentum space, and is convergent for a large class of potentials with or without bound states. As a by-product we obtain simple procedures for the continuation of the physical transition amplitude off the energy shell.